Complete complementary codes (CCCs) play a vital role not only in wireless communication, particularly in multicarrier systems where achieving an interference-free environment is of paramount importance, but also in the construction of other codes that necessitate appropriate functions to meet the diverse demands within today's landscape of wireless communication evaluation. This research is focused on the area of constructing $q$-ary functions for both of {traditional and spectrally null constraint (SNC) CCCs}\footnote{When no codes in CCCs having zero components, we call it as traditonal CCCs, else, we call it as SNC-CCCs in this pape.} of flexible length, set size and alphabet. We construct traditional CCCs with lengths, defined as $L = \prod_{i=1}^k p_i^{m_i}$, set sizes, defined as $K = \prod_{i=1}^k p_i^{n_i+1}$, and an alphabet size of $q=\prod_{i=1}^k p_i$, such that $p_1<p_2<\cdots<p_k $. The parameters $m_1, m_2, \ldots, m_k$ (each greater than or equal to $2$) are positive integers, while $n_1, n_2, \ldots, n_k$ are non-negative integers satisfying $n_i \leq m_i-1$, and the variable $k$ represents a positive integer. To achieve these specific parameters, we define $q$-ary functions over a domain $\mathbf{Z}_{p_1}^{m_1}\times \cdots \times \mathbf{Z}_{p_k}^{m_k}$ that is considered a proper subset of $\mathbb{Z}_{q}^m$ and encompasses $\prod_{i=1}^k p_i^{m_i}$ vectors, where $\mathbf{Z}_{p_i}^{m_i}=\{0,1,\hdots,p_i-1\}^{m_i}$, and the value of $m$ is derived from the sum of $m_1, m_2, \ldots, m_k$. This organization of the domain allows us to encompass all conceivable integer-valued length sequences over the alphabet $\mathbb{Z}_q$. It has been demonstrated that by constraining a $q$-ary function that generates traditional CCCs, we can derive SNC-CCCs with identical length and alphabet, yet a smaller or equal set size compared to the traditional CCCs.

翻译：完全互补码不仅在无线通信（尤其是在多载波系统中实现无干扰环境至关重要）中发挥着关键作用，而且在构建其他需要适当函数以满足当今无线通信评估多样化需求的码字中也具有重要意义。本研究聚焦于构建具有灵活长度、集合规模和字母表的$q$元函数，用于传统完全互补码和频谱零约束完全互补码。我们构建的传统完全互补码长度为$L = \prod_{i=1}^k p_i^{m_i}$，集合规模为$K = \prod_{i=1}^k p_i^{n_i+1}$，字母表大小为$q=\prod_{i=1}^k p_i$，其中$p_1<p_2<\cdots<p_k $。参数$m_1, m_2, \ldots, m_k$（每个均大于或等于$2$)为正整数，而$n_1, n_2, \ldots, n_k$为满足$n_i \leq m_i-1$的非负整数，变量$k$为正整数。为达到这些特定参数，我们在域$\mathbf{Z}_{p_1}^{m_1}\times \cdots \times \mathbf{Z}_{p_k}^{m_k}$上定义$q$元函数，该域被视为$\mathbb{Z}_{q}^m$的真子集，包含$\prod_{i=1}^k p_i^{m_i}$个向量，其中$\mathbf{Z}_{p_i}^{m_i}=\{0,1,\hdots,p_i-1\}^{m_i}$，且$m$的值由$m_1, m_2, \ldots, m_k$之和导出。这种域的组织方式使我们能够涵盖字母表$\mathbb{Z}_q$上所有可能的整数值长度序列。研究表明，通过对生成传统完全互补码的$q$元函数施加约束，可以得到具有相同长度和字母表但集合规模小于或等于传统完全互补码的频谱零约束完全互补码。